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Chapter 9. Phase Diagrams. Phase Diagram Vocabulary. Unary Phase Diagrams – H 2 O. 1 atmosphere. Unary Phase Diagram – Pure Fe. Gibbs Phase Rule (Section 9.17). Tells us how many phases can exist under a given set of circumstances. P+F=C+2 P = number of phases

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chapter 9

Chapter 9

Phase Diagrams

gibbs phase rule section 9 17
Gibbs Phase Rule (Section 9.17)
  • Tells us how many phases can exist under a given set of circumstances.


      • P = number of phases
      • F = number of degrees of freedom – number of variables that can be changed independently of all other variables in the system
      • C=number of components
      • The number two indicates the ability to change temperature and pressure; these are non-compositional variables that affect the phases.
  • Modified Gibbs phase rule
      • Most engineering systems function at a pressure of 1 atmosphere, i.e. we have picked the pressure as one of our degrees of freedom. Therefore,

P+F = C+1

binary isomorphous system
Binary Isomorphous System
  • Two components are completely soluble in each other in both solid and liquid phases
  • Hume-Rothery’s Rules (Section 4.3 text 7th edition)
    • Atomic size difference not greater than 15%
    • Crystal structure is the same for both components
    • Similar electronegativity (i.e. no ionic bonding)
    • Elements have a similar valance
  • Example: Cu-Ni System
    • rCu = 0.128 nm rNi = 0.125 nm
    • Both have a face centered cubic (fcc) structure
    • Electronegativity Cu = 0.19; Ni = 0.18
    • Valance – Cu+ and Cu++; Ni++
cooling curves during solidification
Cooling Curves during Solidification

Solidification occurs at constant temperature while latent heat of fusion is released

cooling curves for a binary isomorphous alloy
Cooling curves for a binary isomorphous alloy
  • Features:
  • Solidus – locus of temperatures below which all compositions are solid
    • Start of solidification during cooling
  • Liquidus – locus of temperatures above which all compositions are liquid
    • Start of melting during heating
modified gibbs phase rule
Modified Gibbs Phase Rule
  • In the liquid or solid phase:
    • P=1, C=2
    • P+F=C+1
    • F=2
    • Both composition and temperature can be varied while remaining in the liquid or solid phase
  • In the L+a region
    • P=2, C=2
    • P+F=C+1
    • F=1
    • If we pick a temperature, then compositions of L and a are fixed
    • If we pick a composition, liquidus and solidus temperatures are fixed



tie line and lever rule
Tie Line and Lever Rule
  • At point B both liquid and a are present
  • WL×R = WS×S





Non-equilibrium cooling results in
    • Cored structure
    • Composition variations in the solid phase as layers of decreasing Ni concentration are deposited on previously formed a phase
    • Solidification point is depressed
    • Melting point on reheat is lowered
  • Homogenization or reheating for extended times at temperature below e’
effect on mechanical properties
Effect on Mechanical Properties

Due to solid solution strengthening, alloys tend to be stronger and less ductile than the pure components.

binary eutectic system

Eutectic temperature

α (solid solution) + β (solid solution)



61.9% Sn


18.3% Sn

97.8% Sn

Binary Eutectic System
  • The two components have limited solid solubility in each other
  • Solubility varies with temperature
  • For an alloy with the Eutectic composition the liquid solidifies into two solid phases
binary eutectic system1
Binary Eutectic System
  • Apply Modified Gibbs Phase Rule
    • Phases present: L, a and b (P=3)
    • Components: Pb and Sn (C=2)
    • P+F=C+1
    • F=0  no degrees of freedom
    • Therefore, three phases can coexist in a binary system only at a unique temperature and for unique compositions of the three phases
    • Upon cooling, there is a temperature arrest during the solidification process (eutectic reaction)
microstructures in the eutectic system
Microstructures in the Eutectic System
  • Depending on the system, eutectic solidification can result in:
    • Lamellar structure – alternating plates
    • Rod-like
    • Particulate
amounts of phases at different temperatures
Amounts of Phases at different temperatures
  • At Teutectic + DT
  • At Teutectic - DT
other reactions in the binary system
Other Reactions in the Binary System
  • Upon Cooling the following reactions are also possible
    • Peritectic L + a b
    • Monotectic L1 L2 + a
    • Eutectoid a b + g
    • Peritectoid a + b g
copper zinc system
Copper-Zinc System
  • Terminal phases
  • Intermediate phases
  • Several peritectics
  • Eutectoid
  • Two phase regions between any two single phase regions
mg pb system
Mg-Pb System
  • Intermediate Compound Mg2Pb
  • Congruently melting

Mg2Pb  L


portion of the ni ti system
Portion of the Ni-Ti System
  • Congruently melting intermediate phase g

g  L


iron carbon system
Iron-Carbon System
  • Reactions on cooling
  • Peritectic

L + d  g

  • Eutectic

L  g + Fe3C

  • Eutectoid

g  a + Fe3C

Cast Iron


iron carbon or iron fe 3 c
Iron-Carbon or Iron-Fe3C
  • In principle, the components of the phase diagram should be iron (Fe) and carbon/graphite (C).
    • Fe and C form an intermediate compound Fe3C, which is very stable
    • There isn’t anything of interest at carbon contents greater than 25 at.% or 6.7 wt.% C.
    • Fe3C is considered to be a component, and the binary phase diagram is drawn using Fe and Fe3C.
  • Names of phases:
    • Ferrite - a iron – bcc structure
    • Austenite – g iron – fcc structure
    • High temperature d iron – bcc structure
    • Cementite – Fe3C
  • Steels have carbon contents <2%, usually <1.2%
  • Cast irons have carbon contents >2%
phase transformations in steels
Phase Transformations in Steels

Eutectoid Composition – 0.76wt% C


Alternating plates (lamellae) of Fe and Fe3C

Austenite  Ferrite + Cementite (at 727ºC upon cooling)

0.76wt.%C 0.022wt.%C 6.7wt.% C

phase transformations in steels1
Phase Transformations in Steels
  • Hypoeutectoid composition <0.76 wt% C
  • Proeutectoid ferrite nucleates and spreads along austenite grain boundaries at T>727ºC
  • Remaining austenite converts to pearlite during eutectoid transformation
phase transformations in steels2
Phase Transformations in Steels
  • Hypereutectoid composition >0.76 wt% C
  • Proeutectoid cementite nucleates and spreads along austenite grain boundaries at T>727ºC
  • Remaining austenite converts to pearlite during eutectoid transformation
phase transformations in steels3
Phase Transformations in Steels



Proeutectoid ferrite


Proeutectoid cementite

effect of alloying elements
Effect of Alloying Elements
  • Addition of an alloying element increases the number of components in Gibbs Phase Rule.
  • The additional degree of freedom allows changes in the eutectoid temperature or eutectoid Carbon concentration